Differential equations worksheet with answers

Free printable Differential equations worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
rr² + br + c?

For the equation y″ + 8y′ + 15y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 8y′ + 15y = 0
r =
2.Easy
t (h)y7 h200

A quantity decays according to dy/dt = −k·y with k > 0. It is 500 at t = 0 and 200 at t = 7 h. Find the rate constant k in h⁻¹.

y(0) = 500y(7) = 200
k = h⁻¹
3.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 3 and y′(0) = 6. Find y(1.5).

y″ + 6y′ + 8y = 0y(0) = 3y′(0) = 6
y(1.5) =
4.Medium
t (min)T (°C)15Ta​12 °C21 °C

A can of soda at 3 °C is taken out of the fridge into a room at 29 °C. After 15 min its temperature is 12 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 21 °C?

T0 = 3 °CTa = 29 °CT(15) = 12 °CT = 21 °C
t = min
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 2y′ + 5y = 5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 5y = 5y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 1y(1) = ?

A system obeys y″ + y′ + 4.25y = 4.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + y′ + 4.25y = 4.25y(0) = 0y′(0) = 0
y(1) =

Answer key

  1. 1. r = −3
  2. 2. k = 0.1309 h⁻¹
  3. 3. y(1.5) = 0.4332
  4. 4. t = 41.61 min
  5. 5. ymax = 1.208
  6. 6. y(1) = 1.115

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